Convex separation from convex optimization for large-scale problems

TL;DR

This paper introduces a memory-efficient convex separation scheme based on Gilbert's algorithm, suitable for large-scale problems, with applications in quantum information theory including nonlocality certification and bounds on quantum states.

Contribution

The authors develop a novel convex separation method that requires minimal memory and is independent of problem dimension, with demonstrated advantages in quantum information applications.

Findings

Outperforms existing linear programming methods in large-scale quantum problems

Certifies nonlocality with up to 42 measurement settings

Provides improved bounds on quantum state visibilities and Grothendieck's constant

Abstract

We present a scheme, based on Gilbert's algorithm for quadratic minimization [SIAM J. Contrl., vol. 4, pp. 61-80, 1966], to prove separation between a point and an arbitrary convex set $S\subset\mathbb{R}^{n}$ via calls to an oracle able to perform linear optimizations over $S$. Compared to other methods, our scheme has almost negligible memory requirements and the number of calls to the optimization oracle does not depend on the dimensionality $n$ of the underlying space. We study the speed of convergence of the scheme under different promises on the shape of the set $S$ and/or the location of the point, validating the accuracy of our theoretical bounds with numerical examples. Finally, we present some applications of the scheme in quantum information theory. There we find that our algorithm out-performs existing linear programming methods for certain large scale problems, allowing us to certify nonlocality in bipartite scenarios with upto $42$ measurement settings. We apply the algorithm to upper bound the visibility of two-qubit Werner states, hence improving known lower bounds on Grothendieck's constant $K_G(3)$. Similarly, we compute new upper bounds on the visibility of GHZ states and on the steerability limit of Werner states for a fixed number of measurement settings.